openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /LocalFlow /MIntegralCurveUniformInvariant.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.LocalFlow.LocalDomainFlow | |
| namespace OAI | |
| noncomputable section | |
| namespace WeakMTWTransport | |
| open Set MeasureTheory Manifold Bundle | |
| open scoped ContDiff Manifold ENNReal NNReal Topology | |
| open Set Filter | |
| open scoped Topology NNReal | |
| open Set Filter | |
| open scoped Topology | |
| open Set Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology | |
| open Set | |
| open scoped Topology | |
| open Set Filter Manifold Bundle ContinuousLinearMap | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter ContinuousLinearMap InnerProductSpace | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open scoped NNReal | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| open Set Filter ContinuousLinearMap | |
| open scoped Topology | |
| open MeasureTheory | |
| open scoped ContDiff ENNReal | |
| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory | |
| open scoped Topology ContDiff Manifold Bundle ENNReal | |
| open Set Filter Manifold MeasureTheory Bundle | |
| open scoped ENNReal ContDiff Topology Manifold | |
| open Set Filter Manifold Bundle ContinuousLinearMap | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff Manifold Bundle | |
| open Set Filter Bundle | |
| open scoped Topology Bundle | |
| open scoped Topology | |
| open Function Manifold Set | |
| section | |
| variable | |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M] | |
| [T2Space M] | |
| lemma exists_isMIntegralCurve_of_uniform_invariant_set [BoundarylessManifold I M] | |
| {v : (x : M) → TangentSpace I x} | |
| (hv : CMDiff 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M))) | |
| {ε : ℝ} (hε : 0 < ε) {K : Set M} | |
| (h : ∀ x ∈ K, ∃ γ : ℝ → M, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)) | |
| (hinv : ∀ (a : ℝ) (γ : ℝ → M), γ 0 ∈ K → | |
| IsMIntegralCurveOn γ v (Ioo (-a) a) → MapsTo γ (Ioo (-a) a) K) | |
| (x : M) (hx : x ∈ K) : ∃ γ : ℝ → M, γ 0 = x ∧ IsMIntegralCurve γ v := by | |
| let s := { a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a) } | |
| suffices hbdd : ¬BddAbove s by | |
| rw [not_bddAbove_iff] at hbdd | |
| rw [exists_isMIntegralCurve_iff_exists_isMIntegralCurveOn_Ioo hv] | |
| intro a | |
| obtain ⟨y, ⟨γ, hγ1, hγ2⟩, hlt⟩ := hbdd a | |
| exact ⟨γ, hγ1, hγ2.mono <| Ioo_subset_Ioo (neg_le_neg hlt.le) hlt.le⟩ | |
| intro hbdd | |
| set asup := sSup s with hasup | |
| obtain ⟨a, ha, hlt⟩ := Real.add_neg_lt_sSup (⟨ε, h x hx⟩ : Set.Nonempty s) (ε := - (ε / 2)) | |
| (by rw [neg_lt, neg_zero]; exact half_pos hε) | |
| rw [mem_ofPred] at ha | |
| rw [← hasup, ← sub_eq_add_neg] at hlt | |
| obtain ⟨γ, h0, hγ⟩ := ha | |
| have hεle : ε ≤ asup := le_csSup hbdd (h x hx) | |
| obtain ⟨γ1_aux, h1_aux, hγ1⟩ := h (γ (-(asup - ε / 2))) | |
| (hinv a γ (h0 ▸ hx) hγ ⟨by linarith, by linarith⟩) | |
| rw [← isMIntegralCurveOn_comp_add (dt := asup - ε / 2)] at hγ1 | |
| set γ1 := γ1_aux ∘ (· + (asup - ε / 2)) with γ1_def | |
| have heq1 : γ1 (-(asup - ε / 2)) = γ (-(asup - ε / 2)) := by simp [γ1_def, h1_aux] | |
| obtain ⟨γ2_aux, h2_aux, hγ2⟩ := h (γ (asup - ε / 2)) | |
| (hinv a γ (h0 ▸ hx) hγ ⟨by linarith, hlt⟩) | |
| rw [← isMIntegralCurveOn_comp_sub (dt := asup - ε / 2)] at hγ2 | |
| set γ2 := γ2_aux ∘ (· - (asup - ε / 2)) with γ2_def | |
| have heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2) := by simp [γ2_def, h2_aux] | |
| simp_rw [Set.mem_Ioo, ← sub_lt_iff_lt_add, ← lt_sub_iff_add_lt, ← Set.mem_Ioo] at hγ1 | |
| simp_rw [Set.mem_Ioo, lt_sub_iff_add_lt, sub_lt_iff_lt_add, ← Set.mem_Ioo] at hγ2 | |
| set γ_ext : ℝ → M := piecewise (Ioo (-(asup + ε / 2)) a) | |
| (piecewise (Ioo (-a) a) γ γ1) γ2 with γ_ext_def | |
| have heq_ext : γ_ext 0 = x := by | |
| rw [γ_ext_def, piecewise, ite_eq_left ⟨by linarith, by linarith⟩, piecewise, | |
| ite_eq_left ⟨by linarith, by linarith⟩, h0] | |
| suffices hext : IsMIntegralCurveOn γ_ext v (Ioo (-(asup + ε / 2)) (asup + ε / 2)) from | |
| (not_lt.mpr <| le_csSup hbdd ⟨γ_ext, heq_ext, hext⟩) <| lt_add_of_pos_right asup (half_pos hε) | |
| apply (isMIntegralCurveOn_piecewise (t₀ := asup - ε / 2) hv _ hγ2 | |
| ⟨⟨by linarith, hlt⟩, ⟨by linarith, by linarith⟩⟩ | |
| (by rw [piecewise, ite_eq_left ⟨by linarith, hlt⟩, ← heq2])).mono | |
| (Ioo_subset_Ioo_union_Ioo le_rfl (by linarith) (by linarith)) | |
| exact (isMIntegralCurveOn_piecewise (t₀ := -(asup - ε / 2)) hv hγ hγ1 | |
| ⟨⟨neg_lt_neg hlt, by linarith⟩, ⟨by linarith, by linarith⟩⟩ heq1.symm).mono | |
| (union_comm _ _ ▸ Ioo_subset_Ioo_union_Ioo (by linarith) (by linarith) le_rfl) | |
| end | |
| open Manifold Bundle | |
| open scoped Manifold Bundle | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] | |
| [I.Boundaryless] | |
| lemma hasMFDerivAt_of_chart_ode {v : (x : M) → TangentSpace I x} | |
| (a : M) {f : ℝ → E} {t : ℝ} | |
| (ht : f t ∈ (extChartAt I a).target) | |
| (hf : HasDerivAt f | |
| (tangentCoordChange I ((extChartAt I a).symm (f t)) a | |
| ((extChartAt I a).symm (f t)) (v ((extChartAt I a).symm (f t)))) t) : | |
| HasMFDerivAt 𝓘(ℝ,ℝ) I ((extChartAt I a).symm ∘ f) t | |
| ((1 : ℝ →L[ℝ] ℝ).smulRight (v ((extChartAt I a).symm (f t)))) := by | |
| let xₜ := (extChartAt I a).symm (f t) | |
| have hft1 := (extChartAt I a).map_target ht | |
| have hft2 := mem_extChartAt_source (I := I) xₜ | |
| let w : E := v xₜ | |
| change HasDerivAt f (tangentCoordChange I xₜ a xₜ w) t at hf | |
| have hder : HasDerivAt ((extChartAt I xₜ ∘ (extChartAt I a).symm) ∘ f) | |
| w t := by | |
| have he : tangentCoordChange I a xₜ xₜ (tangentCoordChange I xₜ a xₜ w) = w := by | |
| rw [tangentCoordChange_comp ⟨⟨hft2,hft1⟩,hft2⟩,tangentCoordChange_self hft2] | |
| rw [← he] | |
| apply HasFDerivAt.comp_hasDerivAt _ _ hf | |
| apply HasFDerivWithinAt.hasFDerivAt (s := range I) _ | |
| (by simp only [ModelWithCorners.range_eq_univ,univ_mem]) | |
| rw [← (extChartAt I a).right_inv ht] | |
| exact hasFDerivWithinAt_tangentCoordChange ⟨hft1,hft2⟩ | |
| refine ⟨(continuousAt_extChartAt_symm'' ht).comp hf.continuousAt,?_⟩ | |
| rw [writtenInExtChartAt,extChartAt_model_space_eq_id,ModelWithCorners.range_eq_univ] | |
| change HasFDerivWithinAt ((extChartAt I xₜ ∘ (extChartAt I a).symm) ∘ f) | |
| ((1 : ℝ →L[ℝ] ℝ).smulRight w) univ t | |
| rw [hasFDerivWithinAt_univ] | |
| convert! hder.hasFDerivAt using 1 | |
| omit [I.Boundaryless] in | |
| lemma contDiffOn_coordinate_vectorField {v : (x : M) → TangentSpace I x} | |
| (hv : ContMDiff I (I.prod 𝓘(ℝ,E)) ∞ | |
| (fun x => (⟨x,v x⟩ : TangentBundle I M))) (a : M) : | |
| ContDiffOn ℝ ∞ (fun y => tangentCoordChange I ((extChartAt I a).symm y) a | |
| ((extChartAt I a).symm y) (v ((extChartAt I a).symm y))) (extChartAt I a).target := by | |
| let e := trivializationAt E (fun x : M => TangentSpace I x) a | |
| have hc : ContMDiffOn I 𝓘(ℝ,E) ∞ (fun x => (e ⟨x,v x⟩).2) e.baseSet := by | |
| intro x hx | |
| exact ((e.contMDiffAt_section_iff hx).mp (hv x)).contMDiffWithinAt | |
| have hmaps : MapsTo (extChartAt I a).symm (extChartAt I a).target e.baseSet := by | |
| intro y hy | |
| change (extChartAt I a).symm y ∈ (chartAt H a).source | |
| simpa only [extChartAt_source] using (extChartAt I a).map_target hy | |
| exact contMDiffOn_iff_contDiffOn.mp (hc.comp (contMDiffOn_extChartAt_symm a) hmaps) | |
| lemma exists_uniform_local_manifold_curves [CompleteSpace E] [HasContDiffBump E] | |
| {v : (x : M) → TangentSpace I x} | |
| (hv : ContMDiff I (I.prod 𝓘(ℝ,E)) ∞ | |
| (fun x => (⟨x,v x⟩ : TangentBundle I M))) (a : M) : | |
| ∃ ε : ℝ, 0 < ε ∧ ∃ U : Set M, U ∈ 𝓝 a ∧ | |
| ∀ x ∈ U, ∃ γ : ℝ → M, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε) := by | |
| let c := extChartAt I a | |
| obtain ⟨r,hr,_,Φ,_,hΦ0,hΦder⟩ := exists_smooth_local_flow_on | |
| (isOpen_extChartAt_target a) (contDiffOn_coordinate_vectorField hv a) | |
| (c.map_source (mem_extChartAt_source a)) | |
| let U := c.source ∩ c ⁻¹' Metric.ball (c a) r | |
| have hUa : U ∈ 𝓝 a := inter_mem (extChartAt_source_mem_nhds a) | |
| ((continuousAt_extChartAt (I := I) (x := a)).preimage_mem_nhds (Metric.ball_mem_nhds _ hr)) | |
| refine ⟨r,hr,U,hUa,?_⟩ | |
| intro x hx | |
| refine ⟨c.symm ∘ (fun t => Φ (t,c x)),?_,?_⟩ | |
| · simp only [Function.comp_apply,hΦ0 _ hx.2,c.left_inv hx.1] | |
| · intro t ht | |
| exact (hasMFDerivAt_of_chart_ode a (hΦder t ht (c x) hx.2).1 | |
| (hΦder t ht (c x) hx.2).2).hasMFDerivWithinAt | |
| end WeakMTWTransport | |
| end | |
| end OAI | |